This is joint work with Barthel, Barrero, Pol and Williamson. Let $\mathcal{G}$ be the category of finite groups and conjugacy classes of surjective homomorphisms. Schwede has defined a corresponding category of globally equivariant spectra, and it is natural to consider the corresponding category of rational spectra, and the Balmer spectrum of its subcategory of compact objects. One can also do the same after replacing $\mathcal{G}$ by a suitable subcategory $\mathcal{U}$. We show that if there is a number $r$ such that all groups in $\mathcal{U}$ can be generated by a set of size $r$, and some other mild and natural conditions are satisfied, then the Balmer spectrum can be identified with the profinite space of isomorphism classes in the profinite completion of $\mathcal{U}$. We also have a number of results that do not require a bound on the number of generators, and in particular, these settle all questions about the case where $\mathcal{U}$ is the category of elementary abelian $p$-groups. The first step is a reduction to algebra, which is fairly straightforward, but the algebra has many interesting and unusual features.

An exotic sphere is a smooth manifold which is homeomorphic, but not diffeomorphic, to a sphere with its standard smooth structure. Many exotic spheres are known to exist, but relatively little is known about their geometric properties. In this talk, I will focus on smooth free circle actions on exotic spheres. These actions are relatively well-understood for exotic spheres which bound parallelizable manifolds, but for exotic spheres which do not bound parallelizable manifolds, only four low-dimensional examples were previously known. I will give an overview of these classical results and then describe recent work, joint with Tilman Bauer, in which we detect infinitely many new examples using recent developments in algebraic topology.

Historically, topological K-theory and its Bott periodicity have been very useful in solving key problems in algebraic and geometric topology. In this talk, we will explore the periodicities of higher real K-theories and their roles in several contexts, including Hill--Hopkins--Ravenel’s solution of the Kervaire invariant one problem. We will prove periodicity theorems for higher real K-theories at the prime 2 and show how these results feed into equivariant computations. We will then use these periodicities to measure the complexity of the RO(G)-graded homotopy groups of Lubin--Tate theories and to compute their equivariant slice spectral sequences. This is joint work with Zhipeng Duan, Mike Hill, Guchuan Li, Yutao Liu, Guozhen Wang, and Zhouli Xu.

Topological Hochschild homology (THH) of the Adams summand (ℓ) of connective complex K-theory (𝑘𝑢) was computed bu Angeltveit, Hill and Lawson using the Bockstein spectral sequence associated to the multiplication by 𝑣₁ in ℓ. However, the morphism of spectral sequences associated to the inclusion ℓ→𝑘𝑢 is not sufficient to lift this computation to the Bockstein spectral sequence associated to the multiplication by 𝑢 in 𝑘𝑢, converging to THH of 𝑘𝑢. Using a novel technique, called gathered spectral sequence, we will show how to leverage the relation between 𝑢 and 𝑣₁ to compute the Bockstein spectral sequence converging to THH of 𝑘𝑢.

I will survey the computations of stable homotopy groups of spheres, recent advances using motivic homotopy theory, their applications to the study of smooth structures on spheres, and the resolution of the final case of the Kervaire invariant problem in dimension 126.

One can think of the stabilisation of an ∞-category as the ∞-category of objects that admit infinite deloopings. For example, the ∞-category of spectra is the stabilisation of the ∞-category of homotopy types. Costabilisation is the opposite notion of stabilisation, where we are interested in objects that allow infinite desuspensions. By connectivity reasons, the costabilisation of the ∞-category of homotopy types is trivial. Chromatic homotopy theory provides us with a filtration of p-local spectra or p-local homotopy types, for each fixed prime number p. The associated graded of this filtration is called the monochromatic layer of height h, for every integer h. In this talk I will show that the costabilisation of the unstable monochromatic layer of positive height h is the stable monochromatic layer of the same height. We can view this as a first step towards understanding chromatic assembly of homotopy types.

The modern perspective on equivariant stable categories is that they are characterized equivalently by the existence of transfers, duality, and the tom Dieck splitting. The purpose of this talk is to explain an analogous characterization of the G-symmetric monoidal structure when G is finite, and a conjectural picture for what happens when G is an infinite compact Lie group. This is joint work with Mike Mandell.

After $K(1)$-localization, Adams's image of $J$ can be regarded as a transfer map. Specifically, it is a transfer map $\Sigma^{-1}KO^{\wedge}_2\to L_{K(1)}S^0$ from the $C_2$-homotopy fixed points to the $\mathbb{Z}_2^\times$-homotopy fixed points of the $2$-complete complex topological $K$-theory. We define transfer maps as Spanier--Whitehead duals to restriction maps in general. For arbitrary heights $n$ and closed subgroups $G$ of the Morava stabilizer group $\mathbb{G}_n$, these transfer maps give analogs of the classical $J$-homomorphism at higher chromatic heights. This is joint work in progress with Guchuan Li.

The Whitehead conjecture predicted the collapse of the homotopy spectral sequence of the symmetric powers filtration. It was proved by Nick Kuhn in 1982, and then one more time in 2013. Around 2007, Kathryn Lesh and I formulated an analogous conjecture involving a filtration of complex K-theory. I will review these results and to extent that time permits will outline a proof of our conjecture. The method also gives a new proof of Kuhn’s theorem. Joint work with Kathryn Lesh, and part of it is also joint with Eva Belmont.

Real algebraic K-theory is a generalization of algebraic K-theory for ring spectra with involution that has applications to quadratic forms and surgery theory of manifolds. Work in progress of Nikolaus—Harpaz—Shah demonstrates that it can be closely approximated by Real topological cyclic homology, such as algebraic K-theory can be closely approximated by topological cyclic homology by the Dundas—Goodwillie—McCarthy theorem. In the non-equivariant, setting recent work of Hahn—Raksit—Wilson, building on work of Bhatt—Morrow—Scholze, demonstrates that topological cyclic homology can be effectively computed using a motivic filtration whose associated graded recovers syntomic cohomology. Syntomic cohomology, and a closely related theory known as prismatic cohomology, are of interest in their own right because of the relation to crystalline cohomology, etale cohomology, and de Rham cohomology. In my talk, I will discuss recent work with Hana Jia Kong and J.D. Quigley that extends the theory of syntomic cohomology to the setting of ring spectra with involution. This also extends a construction of Park in the setting of discrete rings with involution. As an application, we compute the Real syntomic cohomology of certain Real truncated Brown—Peterson spectra and use it to prove a Lichtenbaum—Quillen type result.

Torus localization and the corresponding Bott residue theory is a useful tool for computing classes that arise in enumerative problems. The topological setting due to Atiyah and Bott was revised by Edidin-Graham to a version for the Chow groups. This method was extended to the case of virtual classes by Graber-Pandharipande, and then later using stack methods by Aranha-Khan-Latyntsev-Park-Ravi. Based on results in motivic homotopy theory, one now has available refinements of classical intersection theory, in particular a version that produces invariants in the Witt ring of quadratic forms. However torus localization and its virtual version both fail in the quadratic setting. We will give an overview of the quadratic refinement of intersection theory and the extension of the localization method, the latter using actions by the normalizer of the diagonal torus in SL_2 to replace G_m-actions.

In classical algebra, the Picard group of a commutative ring is invariant under quotient by nilpotent elements. In joint work in progress with Ishan Levy and Ningchuan Zhang, we generalize this to structured ring spectra. Specifically, for a ring spectrum R and an element v in its homotopy group, we establish a vanishing condition under which the map Pic(R/v^{n+1}) -> Pic(R/v^n) is injective, provided R/v admits the structure of an E_1-R-algebra. Applying this to Morava E-theory E_n at height n, we prove that the Picard group of a quotient by a regular sequence in the 0th homotopy groups of E_n is always cyclic of order 2. Using these quotients as input for the profinite descent spectral sequence, we deduce that the Picard group of a K(n)-local generalized Moore spectrum is finite.

Milnor-Witt cycle modules are quadratic analogues of Rost cycle modules, which can be used to define Chow-Witt groups using elementary arithmetic operations on residue fields. We introduce Milnor Witt cycle modules over a base scheme and discuss their relations with the homotopy t-structure on the motivic stable homotopy category. This is a joint work with F. Déglise and N. Feld.

While studying special values of $L$-functions, in the early 80's Beilinson and Lichtenbaum made several conjectures about the existence of a universal cohomology theory with integral coefficients referred to as \emph{motivic cohomology}. In particular, it was conjectured that there should exist an Atiyah--Hirzebruch spectral sequence calculating algebraic $K$-theory from the then conjectural motivic cohomology. For smooth varieties over a field, this spectral sequence was subsequently constructed in multiple ways in work of Bloch, Grayson, Friedlander, Levine, Suslin, and Voevodsky. In this talk I discuss joint work with Shuji Saito where we propose a definition of motivic cohomology over qcqs derived schemes which comes automatically equipped with an Atiyah--Hirzebruch spectral sequence, agreeing with the classical one on smooth varieties. Recent joint work with Shuji Saito--Georg Tamme may also appear.

This is joint work with Sebastian Gant. We work with unital associative commutative rings. A module $P$ over a ring $R$ is said to be stably free of type $(n,t)$ if there exists an isomorphism $P \oplus R^{n-t} \cong R^n$. One may ask whether a general stably free module $P$ of type $(n,t)$ admits a free summand of some rank $s$, i.e., whether one can write $P = Q \oplus R^s$: this family of questions includes the question ``is $P$ necessarily free'' as the case of $s=t$. In 1968, M. Raynaud proved a family of negative results by using geometry and étale cohomology, comparing the problem with that of obstructing certain maps between complex Stiefel manifolds. Among her results is this: a general stably free module of type $(n,n-1)$ does not admit a free summand of rank $2$ unless perhaps if $n$ is a multiple of $24$. We prove positive results: If $R$ contains the field of rational numbers, then a stably free $R$-module $P$ of type $(24m, 24m-1)$ admits a free summand of rank $2$, and if $R$ contains the field of real numbers, then $P$ admits a free summand of rank $3$. The method of proof is to recast the problem as one about Stiefel varieties, which can then be solved by using new results in motivic homotopy, along with comparison to classical topology.

We study the classification of differential graded algebras (DGAs) with polynomial homology Fp[x] (|x|>0) with an eye towards algebraic K-theory computations. This classification question was left open in a work of Dwyer, Greenlees and Iyengar. Our results are surprising. We prove that there are infinitely many DGAs with homology Fp[x] for even |x|>0 and that these DGAs have eqilvanet underlying ring spectra for |x|>2p-3. Through this, we obtain relative algebraic K-theory groups for rings (such as Z[x]/px and Z[C_p^n]) through identifying the DGAs that appear in the motivic pullback squares constructed by Land and Tamme. This work is joint with Markus Land.

Recently, R. Burklund, J. Hahn, I. Levy, and T. Schlank disproved the telescope conjecture. One of the important ingredients in the construction of a counterexample is Levy's generalization of the Dundas-Goodwillie-McCarthy theorem that applies to (-1)-connective ring spectra. There are other unexpected examples of nonconnective ring spectra that behave similarly to connective ones: Z^n-fixed points spectra or the endomorphism ring spectra of generators of derived categories of qcqs schemes. Based on an ongoing work with Ishan Levy, I introduce a new notion of a c-category, which generalizes modules over such ring spectra. We show that such a category admits a finite functorial resolution by categories of modules over connective ring spectra and deduce a DGM theorem for them.